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Cutting stock

Description
A paper mill produces rolls of paper of a fixed width that are subsequently cut into smaller rolls according to the customer orders. The rolls can be cut into NWIDTHS different sizes. The orders are given as demands for each width. The objective of the paper mill is to satisfy the demand with the smallest possible number of paper rolls in order to minimize the losses. Starting with just a basic set of cutting patterns a column generation heuristic is employed to find more profitable cutting patterns. This heuristic solves a second optimization problem (a knapsack problem) that is independent of the main, cutting stock problem.

Further explanation of this example: 'Mosel User Guide', Section 11.2 'Column generation'; Xpress Whitepaper 'Embedding Optimization Algorithms'. Similar problem: 'Applications of optimization with Xpress-MP', Section 9.6 'Cutting steel bars for desk legs'

cutstockgr.zip[download all files]

Source Files

Data Files





cutstock_graph.mos

(!*******************************************************
   Mosel Example Problems
   ======================

   file cutstock.mos
   `````````````````
   TYPE:         Cutting stock problem
   DIFFICULTY:   5
   FEATURES:     MIP problem solved by column generation, 
                 working with multiple problems, using 'sethidden',
                 setting Optimizer and Mosel control parameters
   DESCRIPTION:  A paper mill produces rolls of paper of a fixed width that 
                 are subsequently cut into smaller rolls according to the 
                 customer orders. The rolls can be cut into NWIDTHS different 
                 sizes. The orders are given as demands for each width. 
                 The objective of the paper mill is to satisfy the demand 
                 with the smallest possible number of paper rolls in order 
                 to minimize the losses. 
                 Starting with just a basic set of cutting patterns a 
                 column generation heuristic is employed to find more 
                 profitable cutting patterns. This heuristic solves a 
                 second optimization problem (a knapsack problem) that is
                 independent of the main, cutting stock problem.     
   FURTHER INFO: "Mosel User Guide", Section 11.2 "Column generation";
                 Xpress Whitepaper "Embedding Optimization Algorithms".
                 Similar problem:
                 `Applications of optimization with Xpress-MP', 
                 Section 9.6 `Cutting steel bars for desk legs'
       
   (c) 2008 Fair Isaac Corporation
       authors: Bob Daniel, S. Heipcke, J. Tebboth, 2001, rev. Nov. 2017
  *******************************************************!)

model "Cutting stock"
 uses "mmxprs", "mmsvg"

 parameters
  DATAFILE="cutstock.dat"
  NWIDTHS = 15
 end-parameters

 forward procedure column_gen
 forward function knapsack(C:array(range) of real, 
                           A:array(range) of real, 
                           B:real, 
                           xbest:array(range) of integer,
                           pass: integer): real
 forward procedure print_pat(dj:real, vx: array(range) of integer)
 forward function draw_pat(dj:real, vx: array(range) of integer, iter:integer):boolean

(!
The column generation algorithm requires the solution of a knapsack problem.
We define the constraints of the knapsack problem globally because Mosel 
does not delete them at the end of a function (due to its principle of incremental model formulation).
!)

 declarations
  WIDTHS = 1..NWIDTHS                  ! Range of widths
  RP: range                            ! Range of cutting patterns
  MAXWIDTH: integer                    ! Maximum roll width
  EPS = 0.001                          ! Zero tolerance

  WIDTH: array(WIDTHS) of real         ! Possible widths
  DEMAND: array(WIDTHS) of integer     ! Demand per width
  PATTERN: array(WIDTHS, RP) of integer  ! (Basic) cutting patterns
  PGRAPH: array(WIDTHS) of string      ! Graph for width

  use: dynamic array(RP) of mpvar      ! Rolls per pattern
  soluse: dynamic array(RP) of real    ! Solution values for variables `use'
  Dem: array(WIDTHS) of linctr         ! Demand constraints
  MinRolls: linctr                     ! Objective function
 
  KnapCtr, KnapObj: linctr             ! Knapsack constraint+objective
  x: array(WIDTHS) of mpvar            ! Knapsack variables
 end-declarations

 initializations from DATAFILE
  WIDTH DEMAND MAXWIDTH
 end-initializations
                                       ! Make basic patterns
 forall(j in WIDTHS) PATTERN(j,j) := floor(MAXWIDTH/WIDTH(j))

 forall(j in WIDTHS) do
  create(use(j))                       ! Create NWIDTHS variables `use'
  use(j) is_integer                    ! Variables are integer and bounded
  use(j) <= integer(ceil(DEMAND(j)/PATTERN(j,j)))
 end-do

! Objective: minimize the number of rolls
 MinRolls:= sum(j in WIDTHS) use(j)     

! Satisfy all demands
 forall(i in WIDTHS) 
  Dem(i):= sum(j in WIDTHS) PATTERN(i,j) * use(j) >= DEMAND(i)  

 column_gen                            ! Column generation at top node

 minimize(MinRolls)                    ! Compute the best integer solution
                                       ! for the current problem (including
                                       ! the new columns)
 writeln("Best integer solution: ", getobjval, " rolls")
 write("  Rolls per pattern: ")
 forall(i in RP) write(getsol(use(i)),", ")
 writeln

 svgerase("msg")
 forall(i in RP) svgaddtext("msg", MAXWIDTH+15, 2*i, text(getsol(use(i))))
 svgaddtext("msg", MAXWIDTH+10, 0, "Total: "+getobjval)
 svgrefresh
 svgsave("cutstock.svg")
 svgwaitclose("Close browser window to terminate model execution.", 1)
 
!************************************************************************
!  Column generation loop at the top node:
!    solve the LP and save the basis
!    get the solution values
!    generate new column(s) (=cutting pattern)
!    load the modified problem and load the saved basis
!************************************************************************
 procedure column_gen

  declarations
   dualdem: array(WIDTHS) of real 
   xbest: array(WIDTHS) of integer
   ubnd, zbest, objval: real
   bas: basis
  end-declarations

  setparam("zerotol", EPS)             ! Set Mosel comparison tolerance
  setparam("XPRS_CUTSTRATEGY", 0)      ! Disable automatic cuts
  setparam("XPRS_PRESOLVE", 0)         ! Switch presolve off
  npatt:=NWIDTHS
  npass:=1
  svgrefresh
  forall(j in WIDTHS) do
    PGRAPH(j):="W"+j
    svgaddgroup(PGRAPH(j), "Width "+WIDTH(j))
    svgsetstyle(SVG_FILL,SVG_CURRENT)
    svgsetstyle(SVG_STROKE,SVG_GREY)
    forall(i in 1..PATTERN(j,j))
      svgaddrectangle(10+(i-1)*WIDTH(j),j*2,WIDTH(j),1)
  end-do
  svgaddgroup("msg", "", SVG_BLACK)
  svgsetgraphviewbox(5,-1,MAXWIDTH+30,100)
  svgsetgraphscale(5)

  while(true) do
    minimize(XPRS_LIN, MinRolls)       ! Solve the LP

    savebasis(bas)                     ! Save the current basis
    objval:= getobjval                 ! Get the objective value

                                       ! Get the solution values
    forall(j in 1..npatt)  soluse(j):=getsol(use(j))
    forall(i in WIDTHS)  dualdem(i):=getdual(Dem(i))
                                       ! Solve a knapsack problem
    zbest:= knapsack(dualdem, WIDTH, MAXWIDTH, xbest, npass) - 1.0

    write("Pass ", npass, ": ")   
    if zbest = 0 then
      writeln("no profitable column found.\n")
      break
    else 
      print_pat(zbest, xbest)          ! Print the new pattern
      npatt+=1
      if not draw_pat(zbest, xbest, npatt) then break; end-if
      create(use(npatt))               ! Create a new var. for this pattern
      use(npatt) is_integer

      MinRolls+= use(npatt)            ! Add new var. to the objective
      ubnd:=0
      forall(i in WIDTHS)                 
        if xbest(i) > 0 then
         Dem(i)+= xbest(i)*use(npatt)  ! Add new var. to demand constr.s
         ubnd:= maxlist(ubnd, ceil(DEMAND(i)/xbest(i) )) 
        end-if
      use(npatt) <= ubnd               ! Set upper bound on the new var.

      loadprob(MinRolls)               ! Reload the problem
      loadbasis(bas)                   ! Load the saved basis
    end-if
    npass+=1
  end-do

  writeln("Solution after column generation: ", objval, " rolls, ",
          getsize(RP), " patterns")
  write("   Rolls per pattern: ")
  forall(i in RP) write(soluse(i),", ")
  writeln   
  
 end-procedure

!************************************************************************
!  Solve the integer knapsack problem
!    z = max{cx : ax<=b, x in Z^N}
!  with b=MAXWIDTH, N=NWIDTHS
!
! We reset the knapsack constraints to 0 at the end of this function
! so that they do not unnecessarily increase the size of the main
! problem. The same is true in the other sense: hiding the demand
! constraints while solving the knapsack problem makes life easier
! for the optimizer, but is not essential for getting the correct
! solution.
!************************************************************************
 function knapsack(C:array(range) of real,
                   A:array(range) of real, 
                   B:real, 
                   xbest:array(range) of integer,
                   pass: integer):real

! Hide the demand constraints
  forall(j in WIDTHS) sethidden(Dem(j), true)

! Define the knapsack problem
  KnapCtr := sum(j in WIDTHS) A(j)*x(j) <= B
  KnapObj := sum(j in WIDTHS) C(j)*x(j)

! Integrality condition
  if pass=1 then
   forall(j in WIDTHS) x(j) is_integer
  end-if

  maximize(KnapObj)
  returned:=getobjval
  forall(j in WIDTHS) xbest(j):=round(getsol(x(j)))

! Reset knapsack constraint and objective, unhide demand constraints
  KnapCtr := 0
  KnapObj := 0  
  forall(j in WIDTHS) sethidden(Dem(j), false)

 end-function

!************************************************************************

 procedure print_pat(dj:real, vx: array(range) of integer)
  declarations 
   dw: real
  end-declarations
  
  writeln("New pattern found with marginal cost ", dj)
  write("   Widths distribution: ")
  dw:=0
  forall(i in WIDTHS) do 
    write(WIDTH(i), ":", vx(i), "  ")
    dw += WIDTH(i)*vx(i)
  end-do 
  writeln("Total width: ", dw)
 end-procedure

!************************************************************************

 function draw_pat(dj:real, vx: array(range) of integer, iter:integer):boolean
  if svgclosing then returned:=false; return; end-if
  svgerase("msg")
  returned:=true
  cum:=10.0
  forall(i in WIDTHS, j in 1..vx(i)) do
    svgaddrectangle(PGRAPH(i), cum, 2*iter, WIDTH(i), 1)
    cum+=WIDTH(i)
  end-do  
  forall(j in 1..iter) svgaddtext("msg", MAXWIDTH+15, 2*j, text(soluse(j)))
  svgaddtext("msg", MAXWIDTH+10, 0, "Total: "+dj)

  svgrefresh
  svgpause
 end-function

end-model

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